The chain map \(\tilde{\gamma }_{n}\)  is called a derivation operator. The following theorem due to M. Súarez-Álvarez provides a way for computing the bracket \([\operatorname{HH}^1(\Lambda ),\operatorname{HH}^m(\Lambda )]\)  on Hochschild cohomology. It was first given in [1]} and reformulated by S. Witherspoon in [2]}.
